Chap 1 Complex Numbers 1 Sums and Products

Chap 1. Complex Numbers. 1. Sums and Products. Complex numbers can be defined as ordered pairs (x, y) of real numbers that are to be interpreted as points in the complex plane, with rectangular coordinates x and y, just as real numbers x are thought of as points on the real line. • (x, 0) real number • (0, y) pure imaginary number • It is customary to denote a complex number (x, y) by z , so that x: real part of z Re z = x y: imaginary part of z Im z = y 1

2. Algebraic Properties commutative law associative law distributive law z+0=z 0=(0, 0) additive identity z 1=(1, 0) multiplicative identity For each z, there is a -z such that z+(-z)=0 additive Inverse 3

For any nonzero z=(x, y), multiplicative Inverse There is a less obvious than additive inverse such that • Division by a non-zero complex number if 得到相同結果 4

Other Identities Example. 5

3. Moduli and conjugates It is natural to associate any nonzero complex number z=x+iy with the directed line segment, or vector, from the origin to the point (x, y) that represent z in complex plane. In fact, we often refer to z as the point z or the vector z. y Z 1+Z 2 (x, y) (-2, 1) -2+i 0 x+iy x -Z 2 Z 1 -Z 2 6

• The modulus, or absolute value, of a complex number z=x+iy is defined as length of the vector z. distance between point z and 0 the distance between two points is Z 0 • z x+iy • complex conjugate of z =x+iy is x-iy 7

4. Triangle Inequality geometrically 9

algebraically, Now 10

Example: z on unit circle 2 The triangle inequality can be generalized by mathematical induction to sums … 11

5. Polar coordinates and Euler’s Formula Let r, and be polar coordinates of the point (x, y) that corresponds to a non-zero complex number z=x+iy. since if z=0, the coordinate is undefined. the length of the radius vector for z. has an possible values. Each value of is called an argument of z. and the set of all such values is arg z The principal value of arg z, Arg z, is that unique , s. t. using Euler’s formula then 12

• Two non-zero complex numbers are equal iff 6. Product and Quotients in Exponential From If (1) 13

Moivre’s formula Ex. Find 14

Z 1 Z 2 argument of product (7) Z 2 Z 1 If we know two of these, can find the third. A. If From Expression (1) is a value of B. If If we choose (7) is satisfied. 15

C. Similarly for Z 1 Z 2 Z 1 Then choose Finally Ex: 16

7. Roots of Complex Numbers Suppose z is nth root of a nonzero number . are the nth root of These roots are on the circle and are equally spaced every 17

All of the distinct roots are obtained when k = 0, 1, 2, …, n-1 Let and (і) if (іі) if denote these distinct roots denote the set of nth roots of is a positive real number then denotes the entire set of roots. in (1) is the principal value of arg is referred to as the principal root. 18

Ex. nth roots of unity 1 n=4 ： 1 19

Ex. 2. Find c 1 c 2 2 c 0 20

• Regions in the complex Plane closeness of points to one another • -neighborhood or neighborhood of a given point Z Z 0 • Deleted neighborhood • Interior point A point is said to be an interior point of a set S whenever there is some neighborhood of that contains only points of S. Z 0 S • Exterior point when there exists a neighborhood of containing no points of S 21

• Boundary point all of whose neighborhoods contain points is S and points not in S Boundary = { all boundary points } Ex. is the boundary of and • A set is open if it contains none of its boundary points • A set is closed if it contains all of its boundary points. • The closure of a set S is the closed set consisting of all points in S together with the boundary of S is open is closed and closure of - neither open nor closed. 22

- The set of all complex number is both open and closed since it has no boundary points. • An open set S is connected if each pair of points joined by a polygonal line that lies entirely in S. and in it can be Z 1 open, connected Z 2 1 2 • An open set that is connected is called a domain. (any neighborhood is a domain ) • A domain together with some, none, or all of its boundary points is a region. • A set S is bounded if every point of S lies inside some circle otherwise it is unbounded. ; 23

• A point is said to be an accumulation point of a set S if each deleted neighborhood of contains at least one point of S. - If a set S is closed, then it contain s each of its accumulation points. pf: If an accumulation point of S; were not in S, it would be a boundary (can not be exterior points) but this contradicts the fact that a closed set contains all of its boundary points. Ex: For the set the origin is the only accumulation point. 24